On the Structure of Semimarkov Processes Compared with Chung Processes

Abstract

Semimarkov processes with discrete state spaces are considered without restrictions on their probability laws. They admit states where every visit lasts a positive time even though there may be infinitely many such visits in a finite interval. These are called unstable holding states as opposed to the stable holding states which are encountered in Markov processes. Further, it is possible to have instantaneous states at which the behavior is that at an ordinary instantaneous state of a Chung process, or that at a sticky boundary point, or that at a nonsticky point. To convert such processes to Chung processes, each unstable state is split into infinitely many stable ones, and then a random time change is effected whereby some sets of constancy are dilated, and sojourn intervals are altered to have exponentially distributed lengths.